14_energy_balance.jl

### A Pluto.jl notebook ###
# v0.20.0

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
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        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 3e60b013-98cf-4057-9c96-b627662c85a4
begin
    using DifferentialEquations, Plots, PlutoUI, LinearAlgebra, CSV, DataFrames, Images
end

# ╔═╡ 42085492-ac8c-11eb-0620-adcb307077f1
html"""
<div style="
position: absolute;
width: calc(100% - 30px);
border: 50vw solid #282936;
border-top: 200px solid #282936;
border-bottom: none;
box-sizing: content-box;
left: calc(-50vw + 15px);
top: -200px;
height: 200px;
pointer-events: none;
"></div>

<div style="
height: 200px;
width: 100%;
background: #282936;
color: #fff;
padding-top: 68px;
">
<span style="
font-family: Vollkorn, serif;
font-weight: 700;
font-feature-settings: 'lnum', 'pnum';
"> <p style="
font-size: 1.5rem;
opacity: .8;
"><em>Section 2.1</em></p>
<p style="text-align: center; font-size: 2rem;">
<em> Climate models: greenhouse effect </em>
</p>

<style>
body {
overflow-x: hidden;
}
</style>"""

# ╔═╡ 14195fc4-40e1-4576-973a-69d649fddc02
TableOfContents(title="📚 Table of Contents", indent=true, depth=4, aside=true)

# ╔═╡ 92883ed9-5572-41fd-96c7-190279f90804
md"""
In this lecture we will build a simple climate model that prewdicts the averaged temperature of earth's atmosphere and how it changes in different climates. If you are interested to learn more, you may want to check the "Introduction to Climate Modeling" lecture by former student Henri Drake. [Henri's video](https://computationalthinking.mit.edu/Fall20/lecture20/),
[Henri's original notebook](https://github.com/mitmath/18S191/blob/Fall20/lecture_notebooks/week11/01_energy_balance_model.jl) where he describes the Energy Balance Model.
"""

# ╔═╡ 2ab34e6f-9059-4f33-8abd-df15b0c00b6b
md"""
## Surface temperature and atmospheric $CO_2$
"""

# ╔═╡ 8d2fd4f0-f2f6-4bc8-b753-f34f2abfcc7e
html"""
<iframe width="700" height="394" src="https://www.youtube-nocookie.com/embed/-JIuKjaY3r4" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
"""

# ╔═╡ 041caa91-5449-4de1-8786-56ae3db74bfe
html"""<img src="http://www.euanmearns.com/wp-content/uploads/2014/11/vostok_temperature_co2.png" height=455>"""

# ╔═╡ 21b71237-bfdf-47d1-beb3-eac924c52f6a
md"""
---
---
"""

# ╔═╡ b6c511c8-918d-432c-9756-9641176ff652
md"""
## Model to predict Earth's surface temperature
"""

# ╔═╡ e8a5cf48-f647-4fb3-8c0a-4eae97d03e02
md"""

The goal is to predict the evolution of the average surface temperature of the Earth as a function of external forcing, like the atmospheric concentration of $CO_2$. Let us start by introducing some notation: 

``T=T(t)`` is the Earth's average surface temperature as a function of time $t$ 

Inititally $t=0$, $T(0)=T_0$

We write ``T'`` for ``dT/dt`` and look for an equation of the form:

``T' = f(t)``

where ``f(t)`` represents the external forcing which acts to change the Earth's temperature ``T(t)``
"""

# ╔═╡ 8516277a-2c98-4390-90bb-3a2276ef54e4
md"""
---

#### Solving an ODEs of the form ``T'=f(T,p,t)`` in Julia

0. Define f(T,p,t)
1. ODEProblem(  f, f(0), time_span, parameters)
2. solve
3. plot etc
"""

# ╔═╡ a28f7886-3237-49d0-9c41-84f52aaff2ea
md"""
---

### Earth energy budget
"""

# ╔═╡ 739e914a-9760-4a84-a860-7acadddfc21f
md"""

The energy (heat) budget of a planet without atmsophere is given by the simple equation:

$\begin{align}
\text{\color{brown}{change in temperature}} = & + \text{\color{red}{heating from the Sun's rays}} \newline
& - \text{\color{red}{reflected solar radiation}}\newline
& - \text{\color{blue}{outgoing thermal radiation}}
\end{align}$

where each of these is interpreted as an average over the entire globe.


"""

# ╔═╡ 3a66b385-737e-49e4-b30d-56b055b7bf44
html"""<img src="https://www.solar-facts-and-advice.com/images/Irradiance.jpg" width=625>"""

# ╔═╡ 4b074c56-dc58-4c9c-ae64-7b9ba8d92c20
md"""
---
#### Start with solar heating only

"""

# ╔═╡ ebfeb0b0-d29e-4244-a0e3-a7185f3c8124
md"""

``T' = a ``
$(html"<br>") Solution: ``T(t) = a t + T_0``

"""

# ╔═╡ de443120-f6e1-4fa4-a915-2a316e7fab9d
S_0 = 1366/4; # solar insolation [W/m^2]  (energy per unit time per unit area)

# ╔═╡ 81630f00-2bde-48f2-b8a6-472c732ff8da
C = 355.; # atmosphere and upper-ocean heat capacity [Wyr/m^2/°K]

# ╔═╡ 2251e66d-0188-477c-acf2-14ed3c51e85f
a = S_0/C;

# ╔═╡ b38f98a0-fee8-4746-a136-247e78230f34
begin
	temp₀ = 0; #absolute zero [K]
	timespan = 1000.;
end

# ╔═╡ a1683297-6ad5-4143-bb3a-ff0b6b52ace4
m1 = ODEProblem( (temp, p, t)-> a, temp₀,  (0, timespan) )

# ╔═╡ 3785b312-4228-44e5-992b-90cd23f2bac2
begin
	model1=solve(m1)
	model1.u.-=273.15
	plot(model1,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (-273.15,400),
                          c      = :red,
                          lw     = :3)
	hline!( [temp₀,temp₀] ,c=:white,ls=:dash)
	annotate!( 700, 35+temp₀-273.15, text("Initial Temperature = $(temp₀) K",color=:white))
	title!("Earth's heat budget absorbing solar radiation only")
end

# ╔═╡ 4bf60f27-8db5-42c1-83d8-750bec2060e5
md"""
---

#### Adding reflection of solar radiation
\
``T' = (1 - \alpha)a ``$(html"<br>") Solution: ``T(t) = (1 -\alpha) a t + T_0``

where $\alpha$ is called ablbedo and equal to 30\%.

"""

# ╔═╡ 67153542-28cb-4d8a-ae86-b13be5f84557
alpha = 0.3; # albedo, or planetary reflectivity [unitless]

# ╔═╡ ad973c4b-0162-48fb-b5e0-342d772af800
m2 = ODEProblem( (temp, p, t)-> a * (1-alpha), temp₀,  (0, timespan) )

# ╔═╡ 54801ddc-c1a6-4e82-8264-08c2205d18ed
begin
	model2=solve(m2)
	model2.u.-=273.15
	plot(model2,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (-273.15,400),
                          c      = :red,
                          lw     = :3)
	hline!( [temp₀,temp₀] ,c=:white,ls=:dash)
	annotate!( 700, 35+temp₀-273.15, text("Initial Temperature = $(temp₀) K",color=:white))
	title!("Earth's heat budget absorbing and reflecting solar radiation")
end

# ╔═╡ e2799d05-36a7-4547-8663-90708ce01ce8
md"""
---

#### Adding radiative cooling
\
``T' = (1-\alpha) a - b T^4 ``$(html"<br>")

Comment:  At long times, the temperatures reaches a stable value: 
``T(t) = \left(\frac{(1-\alpha)a }{b}\right)^{1/4}``

"""

# ╔═╡ 3ba79372-aa57-49a8-9d50-146ef8f7d0b3
sigma = 5.670374419*10^(−8); # [W/m^2/K^4]

# ╔═╡ 17800b4d-a879-4270-b39d-c44901f46391
b = sigma/C;

# ╔═╡ 578a80ad-fcf1-4d5c-81e5-205113bbf78d
m3 = ODEProblem( (temp, p, t)-> a * (1-alpha) - b*temp^4, temp₀,  (0, timespan) )

# ╔═╡ 7af3719c-bf0c-4502-a098-66f797f1b6ea
begin
	model3=solve(m3)
	model3.u.-=273.15
	plot(model3,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (-273.15,400),
                          c      = :red,
                          lw     = :3)
	hline!( [temp₀,temp₀] ,c=:white,ls=:dash)
	annotate!( 700, 35+temp₀-273.15, text("Initial Temperature = $(temp₀) K",color=:white))
	title!("Earth's heat budget including outgoing radiation")
end

# ╔═╡ 9fca410f-87bd-405d-8336-3ae109ee60f9
md"""
---

#### Adding an atmosphere
\
``T' = (1-\alpha) a - (1-\epsilon/2)b T^4 ``$(html"<br>")

Comment:  At long times, the temperatures reaches a stable value: 
``T(t) = \left(\frac{(1-\alpha)S }{(1-\epsilon/2)b}\right)^{1/4}``

"""

# ╔═╡ de777b52-8ff7-4fc9-bdbc-17d64abf8926
T₀=273.15-18;

# ╔═╡ ae54e5a6-f4ab-4196-845e-612e7c7ebb5c
md"""

epsilon = $(@bind epsilon Slider(0:.01:1, show_value=true, default=0) )

"""

# ╔═╡ 3aa13ecd-9d4a-4b41-b15f-f85e2b8f8bf6
m4 = ODEProblem( (temp, p, t)-> a * (1-alpha) - (1-epsilon/2)*b*temp^4, T₀,  (0, timespan) )

# ╔═╡ 2d3740ae-bbeb-49a5-a51e-0b5079e68592
begin
	model4=solve(m4)
	model4.u.-=273.15
	plot(model4,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (-30,35),
                          c      = :red,
                          lw     = :3)
	hline!( [(a*(1-alpha)/b)^(1/4)-273.15,(a*(1-alpha)/b)^(1/4)-273.15] ,c=:white,ls=:dash)
	annotate!( 700, -7+T₀-273.15, text("Initial Temperature = $(T₀-273.15) C",color=:white))
	title!("Earth's temperature with an atmosphere")
end

# ╔═╡ 3c3f7997-ca76-4536-8331-634fff7fbcb7
md"""
---

#### Adding carbon to present day atmosphere
\
``T' = (1-\alpha) a - (1-\epsilon(CO_2)/2)b T^4 ``$(html"<br>")


"""

# ╔═╡ 87a384f1-1679-4f1a-aaad-ac85c117b959
T₁=((1-alpha)*a/((1-0.745/2)*b))^(1/4)

# ╔═╡ 7a6019d9-41af-4e5c-80e6-25dc52ac1819
eps(CO₂,T) = 0.745+0.02 * log(CO₂/280)/log(2)+0.01*(T-T₁)

# ╔═╡ ce1792bc-36d2-49b2-bd92-351e09bb3a4f
md"""

CO₂ = $(@bind CO₂ Slider(280:1:560, show_value=true, default=280) )

"""

# ╔═╡ a61b1671-ad76-4a34-8d63-3f687fffe836
m5 = ODEProblem( (temp, p, t)-> a * (1-alpha) - (1-eps(CO₂,T₁)/2)*b*temp^4, T₁,  (0, timespan) )

# ╔═╡ 819431a7-cdcd-430a-a89a-2db90eb22e7c
begin
	model5=solve(m5)
	model5.u.-=273.15
	plot(model5,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (12.4,17),
                          c      = :red,
                          lw     = :3)
	hline!( [T₁-273.15,T₁-273.15] ,c=:white,ls=:dash)
	annotate!( 700, -.5+T₁-273.15, text("Initial Temperature = $(round(10*(T₁-273.15))/10) C",color=:white))
	title!("Earth's temperature deepndence on CO₂")
end

# ╔═╡ ee9724f1-5414-4c3d-85b9-e394afea2e72
md"""
---

#### Adding water vapor feedback
\
``T' = (1-\alpha) a - (1-\epsilon(CO_2,T)/2)b T^4 ``$(html"<br>")


"""

# ╔═╡ 0b6e8dd2-a060-4ed9-aa03-de48e3c7c3e1
m6 = ODEProblem( (temp, p, t)-> a * (1-alpha) - (1-eps(CO₂,temp)/2)*b*temp^4, T₁,  (0, timespan) )

# ╔═╡ 5b153580-78ee-4a73-b936-507b3b39f02c
begin
	model6=solve(m6)
	model6.u.-=273.15
	plot(model6,       legend = false, 
		 background_color_inside = :black,
		                  xlabel = "years",
	                      ylabel = "Temperature °C",
                          ylim   = (12.4,17),
                          c      = :red,
                          lw     = :3)
	hline!( [T₁-273.15,T₁-273.15] ,c=:white,ls=:dash)
	annotate!( 700, -.5+T₁-273.15, text("Initial Temperature = $(round(10*(T₁-273.15))/10) C",color=:white))
	title!("Earth's temperature with an atmosphere")
end

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[[deps.Tricks]]
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version = "3.6.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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